The psychology of mathematics

[This is a condensed version of a paper written by one of the present bloggers (Borwein). For the full article, with references, see]

Some years ago, my brother Peter surveyed other academic disciplines. He discovered that students who bitch mightily about calculus professors still prefer the relative certainty of how-and-what we teach-and-assess to the subjectivity of a creative writing course or the rigors of a physics or chemistry laboratory course. Similarly, while I have met my share of micro-managing Deans–who view mathematics with disdain when they look at the size of our research grants or the infrequency of our patents–I have encountered more obstacles to mathematical innovation within than without the discipline.

Why do we produce so many unneeded results? In addition to the obvious pressure to publish and requirements to have something to present at the next conference, I suspect Irving Biederman’s observations below plays a significant role: “While you’re trying to understand a difficult theorem, it’s not fun,” said Biederman, professor of neuroscience in the USC College of Letters, Arts and Sciences, “But once you get it, you just feel fabulous.” The brain’s craving for a fix motivates humans to maximize the rate at which they absorb knowledge, he said. “I think we’re exquisitely tuned to this as if we’re junkies, second by second.”

Sometimes we sit firmly and comfortably in the sciences, sometimes we are–as the Economist recently noted–the most inaccessible of the arts. Why should the non-mathematician care about things of this nature? The foremost reason is that mathematics is beautiful, even if it is, sadly, more inaccessible than other forms of art: “a supreme beauty–a beauty cold and austere” in Russell’s terms and sometimes we sit or feel we sit entirely alone. On the other hand, the human genome project, the burgeoning development of financial mathematics, finite element modeling, Google and much else has secured the role of mathematics within modern science and technology research and development as “the language of high technology.”

One of the epochal events of my childhood as a faculty brat in St. Andrews, Scotland was when C. P. Snow (1905-1980) delivered an immediately influential 1959 Rede Lecture in Cambridge entitled “The Two Cultures.” Snow wrote:

A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics, the law of entropy. The response was cold: it was also negative. Yet I was asking something which is about the scientific equivalent of: `Have you read a work of Shakespeare’s?’

I now believe that if I had asked an even simpler question – such as, What do you mean by mass, or acceleration, which is the scientific equivalent of saying, ‘Can you read?’ – not more than one in ten of the highly educated would have felt that I was speaking the same language. So the great edifice of modern physics goes up, and the majority of the cleverest people in the western world have about as much insight into it as their Neolithic ancestors would have had.

I doubt I have ever met a scientist who had not read (or at least watched on BBC) some Dickens, who never went to movies, art galleries or the theatre. It is, however, socially acceptable to be a scientific ignoramus or a mathematical dunce. It is largely allowed to boast “I was never any good at mathematics at school.” I was once told exactly that–in soto voce–by the then Canadian Governor General during a formal ceremony at his official residence in Ottawa.

As Underwood Dudley has commented, no one apologizes for not being good at geology in school. Most folks understand that failing “Introduction to Rocks” in Grade Nine does not knock you off of a good career path. The outside world knows several truths: mathematics is important, it is hard, it is usually poorly taught in school, and the average middle-class parent is ill-prepared to redress the matter. I have become quite hard-line about this. When a traveling companion on a plane starts telling me that “Mathematics was my worst subject in school.” I will reply “And if you were illiterate would you tell me?” They usually take the riposte fairly gracefully.

Consider two currently popular TV dramas, Numb3rs (mathematical) and House (medical). A few years ago a then colleague, a distinguished pediatrician, asked me whether I watched Num3rs. I replied “Do you watch House? Does it sometimes make you cringe?” He admitted that it did but he still watched it. I said the same was true for me with Numb3rs, that my wife loved it and that I liked lots about it. It made mathematics seem important and was rarely completely off base. The lead character, Charlie, was brilliant and good-looking with a cute smart girl friend. The resident space-cadet on the show was a physicist not a mathematician. What more could one ask for? Sadly for many of our colleagues the answer is “absolute fidelity to mathematical truth in every jot and title.” No wonder so many of us make a dog’s-breakfast of the opportunities given to publicize our work!

I once wrote “It is certainly rarer to find a mathematician under thirty who is unfamiliar with at least one of Maple, Mathematica or Matlab, than it is to one over sixty five who is really fluent. As such fluency becomes ubiquitous, I expect a re-balancing of our community’s valuing of deductive proof over inductive knowledge.”

At a more fundamental level, I see the discipline boundary as being best determined by answering the question as to whether the mathematics at issue is worth doing in its own right. If the answer is “yes” then it belongs in the discipline; if not then, however useful or important the outcome, it does not. The later would, for example, be the case of a lot of applied operations research, a good deal of numerical modeling and scientific computation, and most of statistics. All significant mathematics should be nourished within mathematics departments, but there are many important and useful applications that do not by that measure belong.

We have a lot to catch up with. We have too few prizes. We are insufficiently adept at boosting our own cases for tenure, promotion or for prizes. We are frequently too honest in reference letters. We are often disgracefully terse–unaware of the need to make obvious what is for us obvious. I have seen a Field’s medalist recommend a talented colleague for promotion with the one line letter “Anne has done some quite interesting work.” Leaving aside the ambiguity of the use of the word “quite” when sent by a European currently based in the United States to a North American promotion committee, it is pretty lame when compared to a three page letter for an astrophysicist or chemist which almost always tells you the candidate is the top whatever-it-is in the field.

I’m not encouraging dishonesty, but it is necessary to understand the ground rules of the enterprise and to make some attempt to adjust to them. When a good candidate for a Rhodes Scholarship turns up at ones office, it should be obvious that a pro forma “Johnny is smart and got an A+ in my advanced algebraic number theory class. You should give him a Rhodes scholarship.” is inadequate. Yet the only letters of that kind that I’ve seen in Rhodes scholarship dossiers have come from mathematicians.

I became a mathematician largely because it satisfied three criteria. (i) I found it reasonably easy (ii) I liked understanding or working out how things function, but (iii) I was not much good with my hands and had limited physical intuition, I really disliked pipettes. That left mathematics. I have had several students whom I can not imagine following any other life path but I was not one of those. I would I imagine have been happily fulfilled in various careers of the mind; say as an historian or an academic lawyer. But I became a mathematician. It has been and continues to be a wonderful life.

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